2.1

Quantitative expression of compound expression model

Reference [4] integrated the cone model and the direction relationship matrix model to establish a compound expression model describing the qualitative direction relationship, but did not quantify the model. This article establishes the quantitative calculation method of the compound expression model based on the grid. First, the space is divided into 25 partitions by referring to the straight line where the smallest outer rectangle side of the target is located {NW, NNW, N, NNE, NE, WNW, TL, T, TR, ENE, W, L, Same, R, E, WSW, BL, B, BR, ESE, SW, SSW, S, SSE, SE}, corresponding to the 25 elements of the 5×5 order matrix, and then the space object. The area is divided into fixed-size grids, and the values of 25 elements are the proportions of the number of grids in the partition, as shown in equation (1).

When calculating the total number of grids of a certain spatial object, those that are less than one full grid are also regarded as one grid; when calculating the number of grids in the direction slice, only those with more than half a grid are regarded as one grid. The compound expression model divides the space around the reference target more finely, which can avoid the ambiguity of the direction relationship caused by the size, distance and shape of the source target.

2.2.

Decomposition principles of composite spatial objects

When expressing and calculating the direction relationship of the composite spatial object, the direction relationship matrix of the compound expression model is introduced in the following situations, depending on whether or not the reference target A and the source target B are composite spatial objects.

2.2.1

The Reference Target Is a Composite Spatial Object.

When the reference target A is a composite spatial object composed of m single targets, the m single targets are used as the reference targets, and the directional relationship matrix between each single target and the source target is calculated, and then the m single targets are located. The number of grids occupied is weighted proportionally, and the direction relationship matrix between m single targets and the source target is summed, and finally the direction relationship matrix between the composite spatial object and the source target is obtained, namely

where, DIR(A_i, B) is the direction relationship matrix between the i-th single target in the composite reference target and the source target p_i, which is the ratio of the number of grids occupied by the i-th single target in the composite reference target.

4.1

Representation of the direction relation matrix of the composite reference target

First, in Figure 3, the composite spatial object A is taken as the reference target, and the composite spatial object B is taken as the source target, which is represented by the direction relationship matrix based on the quantification of the compound expression model in this article. As shown in Figure 4, in order to decompose according to the idea of Section 2.2.3, it is required to divide the composite reference target A into 3 single reference targets ((b)-(d) in Figure 4), and then obtain the source target and each single reference separately. The matrices of directional relationships between targets are as follows.

Figure 4.

Division of composite reference target A.

It is required to obtain the direction relationship matrix between each single reference target and the composite source target, as shown in Figure 5, synthesize the cone model and the direction relationship matrix model to establish a compound expression model reference system, and calculate a single reference system based on the number of grids in each direction slice. The direction relationship matrix between the reference target A1 and the source target B is as (4), the direction relationship matrix between the single reference target A2 and the source target B is as (5), and the direction relationship matrix between the single reference target A3 and the source target B is as (6).

Figure 5.

The establishment of a compound expression model reference system for each single target.

According to the grid proportion weight occupied by each single reference target, the direction relationship matrix between the composite reference target A and the composite source target B is calculated as equation (7):

In the same way, the direction relation matrix of Figures 3b and 3c can be obtained as equations (8) and (9), respectively:

4.2

Calculation of similarity of composite reference target direction relationship

In order to compare and analyze the similarity between the large-scale map and the small-scale map, it is necessary to calculate the directional similarity of Figures 3a and 3b, and calculate the directional similarity of Figures 3a and 3c. Let’s take the calculation of direction similarity in Figures 3a and 3b as an example.

Step 1: The matrix difference ΔDir of the direction relation matrix is calculated, and the equation (10) is obtained.

Step 2: A direction relationship matrix conversion problem table is constructed based on ΔDir, the direction slice corresponding to the positive element value is written into the left column of the box, and the corresponding positive element value is put in the right column of the box; the slices are written into the upper row of the box, and the absolute value of the corresponding negative element value is written into the lower row of the box; for the distance between the other squares in the table in the writing direction of the slices, please refer to Figure 2, ΔDir. The conversion problem table is shown in Figure 6:

Figure 6.

Table of transformation problems of directional relation matrix.

Step 3: According to the calculation process of Figure 7, starting from the smallest distance value of the direction piece, the value of the positive and negative elements in the row and column is compared with the distance value of the direction piece as the intersection point, and then the smaller row (or column) is deleted. The element value outside the box becomes 0, and the intersecting column (or row) is subtracted from the element value corresponding to the deleted row (column), and the next smallest direction slice is processed in the same way. As shown in Figure 7, starting from the distance value of 2, the element value 4/342 in the corresponding row and column is smaller, the element value of this column becomes 0, and the element value 9/342 of the intersecting row is obtained by subtracting 4/342.5/342.

Figure 7.

The conversion process of the direction relation matrix conversion problem table.

Step 4: The directional distance is calculated, the distance of the directional slice is multiplied where the transition occurs by the variable next to it, and then it is summed up.

Step 5: The similarity of the direction relation matrix is calculated, d_max represents the maximum distance between the direction slices, the value of which is 8.

In the same way, the directional similarity between Figures 3a and 3c is calculated to obtain

The calculation results in this article also show that the directional similarity of Figures 3a and 3b is higher than the directional similarity of Figures 3a and 3c, which is consistent with human subjective visual cognition.